- For any subset $S$ of $k$ dragons such that every dragon is in $S$ or a neighbor of a dragon in $S$, Minnie can guarantee at most $k$ dragons survive, just by choosing the dragons in $S$ repeatedly until no others are left. - For any subset $S$ of $k$ dragons such that no two dragons in $S$ are neighbors or have common neighbors, Max can guarantee at least $k$ dragons survive. He does so by never destroying a dragon in $S$ unless he has no other choice. In this way, either each dragon in $S$ survives, or it is destroyed by a dragon who is thus isolated from all other dragons and survives. These dragons don't overlap since no dragons in $S$ have common neighbors. The following diagram shows the answer is 4 for #1. (same subset for both) .O.. ...O O... ..O. The following diagram shows the answer is at least 6 for #2. O...O ..O.. ..... O...O ..O.. Label each dragon with $(x,y)$ for $1 \le x,y \le 5$. Minnie can guarantee 6 as follows. - Have dragon $(3,3)$ destroy all its neighbors. - Choose the dragon in $(1,1)$. WLOG, he destroys $(2,1)$. - Have dragon $(1,2)$ destroy all its neighbors. - Choose the dragon in $(1,5)$. The dragon not chosen is then made to destroy all its neighbors. This means one dragon out of $(1,4), (2,4), (1,5), (2,5)$ still lives. - Minnie has dragons $(4,1), (5,3), (4,5)$ destroy all of their neighbors. This leaves only 6 dragons alive.